2.1 Pure ice

The relaxation spectrum of pure polycrystalline ice shows that ice is a dielectric solid containing polar molecules that are capable of orientation in an applied A.C. electric field. The motion of the molecular dipoles in the A.C. field is described by an over-critically damped oscillation with a relaxation time τ. Polarization in the ice results in a complex relative permittivity ϵ, which is given by the well-known Debye dispersion relation, Equation (2) in Section 2.3. Ice has several unusual features: for example, the relative permittivity of pure ice at constant temperature can be described by a single relaxation time τ _i, while most solids show a wide distribution of τ; and the long _{τ i ≈ 10⁻⁴} sec. gives a large dispersion in the audio- and low radio-frequency range.

The relaxation time of ice τ _i is influenced by temperature, impurities and stress. The effect of temperature is to produce a thermal agitation of the dipoles which hinders their ordered orientation in the field. When the temperature is lowered, there is an increase in τ _i, since the ability of the dipoles to align in the external field is reduced at low temperatures. τ _i is related to the temperature T through the Boltzmann factor,

where the characteristic constants of a dipole in ice are the time constant, τ ₀ = 5.3×10⁻⁶sec., and the activation energy for dipole orientation,W _i = 13.25 kcal. mole⁻¹, (after Reference Ozawa and KuroiwaOzawa and Kuroiwa, 1958). A distribution of relaxation times is possible, if the temperature through the ice is not uniform.

Impurities give an increase in the electric conductivity of ice. The presence of cracks, bubbles and other imperfections in ice alter the physical structure of ice and hence its response to an A.C. field. Impurities tend to shorten τ _i (Reference GränicherGränicher and others, 1957; Reference BrillBrill, 1957). Mechanical stress also has been found to shorten τ _i (Westphal, quoted by Reference EvansEvans, 1965).

2.2 Snow

The relaxation spectrum of snow is similar to that of ice, but it can show wide variations due to the effect of other factors present in snow. The spectrum is affected by differences in the snow density ρ _s, and structure, as well as by temperature and impurities. Snow changes its internal structure with time, so that not only does ρ _s change but also the arrangement and the linkage between the individual snow crystals. Although the relaxation time of snow τ _s of known density ρ _s is shorter than τ _i of ice (Reference YosidaYosida and others, 1958), there is no obvious way in which τ _s can be related to ρ _s, except to say that τ _s becomes longer with an increase in ρ _s and approaches τ _i. There is no definite relationship between τ _s and temperature T, except that τ _s increases with a decrease in T for a given snow sample. Impurities in snow tend to shorten τ _s. A distribution in τ _s is possible if a temperature or density gradient exists through the snow.

2.3 Theory and measurement of τ

A polar dielectric solid, with a complex relative permittivity ϵ = ϵ′−jϵ″, has a frequency dependence given by the Debye dispersion relation,

The relative permittivity ϵ′ at a frequency ω is

where ϵ ₀ and ϵ _∞ are the relative permittivities in the cases of ω→0 and ω→∞ respectively.

The loss factor ϵ″ is

ϵ′ is associated with the polarization of charge in the dielectric, and ϵ″ takes account of energy dissipation in the dielectric. The relaxation time T can be derived from the maximum of the ϵ″−ϵ′ graph. This follows since all points obeying the Debye relation lie on a semicircle in the ϵ″−ϵ′ graph. This is a Cole-Cole semicircle, diameter (ϵ ₀−ϵ _∞) and centred at (ϵ ₀+ϵ _∞)/2 on the ϵ′ axis: representative Cole-Cole diagrams are shown in Figures 1 and 2. The ϵ″ and ϵ′ values measured at different ω form the Cole-Cole semicircle, and the maximum ϵ″ value occurs at an angular frequency ω _m, which satisfies ωτ = 1 in Equation (4). The maximum frequency, f _m in Hz. or c. sec.⁻¹ corresponds to the maximum energy absorption and power loss in the dielectric, and the relaxation time τ is given by

Fig. 1. A Cole-Cole diagram of compact snow of density 0.4 g. cm.⁻³ at −9.0°C. (after Reference Ozawa and KuroiwaOzawa and Kuriowa, 1958)

Fig. 2. A Cole-Cole diagram for wet snow of density 0.4 g. cm.⁻³ at 0°C. (Curve c), and for the. same snow sample at −9.0° C. (Curve d) (after Reference Ozawa and KuroiwaOzawa and Kuriowa, 1958)

Equation (5) is used to define the relaxation times, τ _i or τ _s, in the present work.

When there is a distribution in τ, the centre of the Cole-Cole semicircle is depressed below the ϵ′ axis, and α, the angle expressed in radians, between the ϵ′-axis and the line joining the centre of the circle to ϵ _∞, is the spreading factor of τ (Reference BöttcherBöttcher, 1952). τ again is defined by Equation (5).

When impurities are present, a Cole-Cole diagram shows high ϵ″ values at low frequencies. This is due to ionic conductivity effects. Impurity ions can give the dielectric an apparent conductivity, σ in mho m.⁻¹, derived from the apparent parallel leakage resistance across the capacitance being observed by the A.C. field. σ differs from the ohmic conductivity a and the associated Joule heating losses in a D.C. field, since there are additional power losses, expressible by a loss factor ϵ‴, in the A.C. field from dipole reversal in the dielectric. The extra dissipation ϵ‴ is related to σ by

where ϵ ₀ is the relative permittivity in vacuum. Since ϵ‴ ≈ ϵ″ at low frequencies ϵ‴ can contribute appreciably to the power loss, but at high frequencies ϵ‴≪ϵ″ and the effect of ϵ‴ is negligible. A Cole-Cole semicircle therefore still can be constructed through the measured ϵ″ and ϵ′ values at high frequencies, and T obtained from the f _m value at the maximum value of ϵ″ of the semicircle.

It has been assumed that impurities, in general, do not affect the dielectric properties of the ice itself with the result that the ϵ _∞,ϵ ₀ and f _m values of ice are unaltered by the presence of impurities. This however, is not strictly true (for example see Section 2.1).

There are wide variations possible in the internal structure of ice. Cracks in various directions, air spaces, the orientation of individual crystals, stress, all change the value of ϵ _o and hence the diameter of the Cole-Cole semicircle with a resultant nett decrease in τ. τ can be obtained also from the relaxation spectrum, or frequency dependence of ϵ′ and ϵ″. The spectrum shows that, while ϵ′ decreases monotonically with increasing frequency, ϵ″ reaches a maximum at a frequency f _m, given by Equation (5). However, if ϵ‴ > 0 at the frequency f _m the ϵ″ maximum is less definite or even absent, so that τ can not be determined by this method.

2.4 Cole-Cole diagrams

Several Cole-Cole diagrams relevant to the present work are shown in Figures 1 and 2. Figure 1 shows a Cole- Cole diagram of compact snow, ρ _s = 0.4 g. cm.⁻³ , T = −3.0°C., obtained by Reference Ozawa and KuroiwaOzawa and Kuroiwa (1958). Curve A is the Cole-Cole semicircle measured immediately after snow was packed into the capacitor used for the ϵ′ and ϵ″ measurements, and curve b is for the same snow after it had been kept at T = 3.0°C. for 24 hr. ϵ _∞ remained constant, but ϵ ₀ increased due to the changes in crystalline structure and the formation of “ice bridges” between the individual snow crystals. Impurity ions were responsible for the rapid rise in ϵ″ at low frequencies. The relaxation time τ _s became longer with time and tended towardsτ _i.

Figure 2, curve c, shows the Cole-Cole diagram for wet snow, ρ _s = 0.38 g. cm.⁻³, T = 0°C., that had been studied when the air temperature was 1.8°C. above the freezing point, so that the snow was wet and contained a small percentage of free liquid water. Curve c was obtained after the snow just had been put in the capacitor, and curve n when the snow in the capacitor was placed in a cold chamber maintained at −9.0°C. In both curves c and D the low frequency, ϵ‴ loss contribution was very large. ϵ _∞ and ϵ ₀ were decreased by the freezing and the lowering of the snow temperature. It was found also that τ _s for wet snow at 0°C. was less than τ _s for dry compact snow at a temperature just below 0°C.

3.1 Theory of the experiment

When an A.C. electric field is applied to electrodes placed on top of the surface or within the bulk of a glacier, the electrode system will act as a capacitor with a semi-infinite lossy dielectric between the electrodes. When the electrodes are placed on the surface, the measured values of the capacitance C and the loss tangent D will depend on the dielectric properties of the glacier.

Two wires were placed on the surface, and C and D measured using an A.C. bridge. The electrode system was assumed, following Evans (private communication), to represent a lossy capacitor with an equivalent parallel capacitance C _p given by

where C _s is the series capacitance of the system.

It is assumed to a first approximation also that

where G is a geometrical factor dependent on the electrode and array dimensions. It follows that

and, since D = ϵ″/ϵ′, that

The Cole-Cole diagrams will give f _m or τ

3.2 Equipment and its use

The electrode system was placed on the surface of the glacier. Two thin insulated parallel wires, radius 4×10⁻⁴ m., were connected at one end to form a П-shaped array, with maximum wire separation b ≈ 200 m. and maximum length l ≈ 350 m. The parallel wires were connected by wires of the same material to an A.C. bridge and oscillator situated half-way between the wires.

An A.C. voltage, V ≈ 1 or 2 V., was supplied by a Levell transistor R.C. oscillator type TG 150 DM, frequency range 1.5 Hz. to 150 kHz., powered by two 9 V. dry batteries. The A.C. bridge was a Marconi TF 2700 transistorized bridge, powered by a 9 V. battery, on which C _s and D could be measured. The bridge usually was easy to balance, although above 20 kHz., the upper limit for accurate bridge measurements, the null point could be arbitrary. Modifications were made to the A.C. bridge to include an intermediate D × 0.1 range between the D × 0.01 and D × 1 ranges available on the bridge. All the measured loss tangent values D′ had to be scaled by the frequency f, in kHz. at which a measurement was made, to give the correct loss tangent D = D′f, since the D values of the bridge were graduated for 1 kHz.

Measurements of C _s and D were made for different frequencies in the range 250 Hz. to 60 kHz. with the first array of small separation, and repeated with different lengths. The wires were held in place at the ends and corners by wood or metal pegs, and additional lengths of wire were added using plastic connectors. The separation then was increased, and the procedure repeated. Correction runs for the capacitance of the side arms were made for large b values.

3.3 Glaciers under study

The measurements were carried out on two temperate glaciers; the first being in the Sukkertoppen region of West Greenland, and the second above Chamonix in the French Alps.

The “Big Array” glacier (lat. 65° 45′ 30″ N., long. 52° 18′ W.) lies south of Evighedsfjord in West Greenland and about 50 km. from the open sea. During 1–9 August 1965, the runs were carried out at 855 m., an altitude about 200 m. above the firn line and in the “soaked facies” region of the glacier. Since the air temperature was above freezing point, the upper layers of the glacier consisted of wet snow at 0°C., with a relatively high percentage of free liquid water from summer melt water, and the 7 cm. rain precipitated during 1–4 August. The measured 0°C. snow temperature was in agreement with the estimated snow temperature at a TO m. depth at the altitude and latitude of the “Big Array” glacier (Reference Mock and WeeksMock and Weeks, 1966). The “Big Array” glacier may be classified as “temperate” in its accumulation area, a conclusion in agreement with the “temperate” classification of the Sukkertoppen Iskappe (lat. 66° 15′ N., long. 52° W.), 40 km. to the north, based on temperature measurements made by Reference BullBull (1963), Reference RundleRundle (1965) and Reference LoeweLoewe (1966). The lower layers of the glacier in the accumulation area also are assumed to be close to 0°C.

Snow from a pit dug to a depth of 2 m. gave a rather high average ρ _s = 0.77±0.03 g. cm.⁻³ in the rain-saturated upper snow layer at 0°C. of the “Big Array” glacier. The estimated mean annual snow accumulation was 16.5± 1.0 g. cm.⁻² over the period 1958–64, an accumulation comparable to that of the central area of the Sukkertoppen Iskappe (Reference BullBull, 1963).

The glacier at Chamonix was the plateau of the Vallée Blanche at 3550 m., close to the Col de Midi and “Les Cosmiques”, the C.N.R.S. laboratory on the Aiguille du Midi. Runs were made in fine weather during 1–4 April 1966. The dry cold upper layers of the glacier were below 0°C. due to the penetration of the winter cold wave into the surface layers of wind-blown consolidated powder snow and underlying névé; the air temperature was less than 0°C. Below the region of the cold wave the glacier was at the pressure melting temperature near 0°C. Certain facts are known about the glacier. A névé layer, thickness about 30 m., over 150 m. of ice should cover most of the plateau, according to the seismic refraction results of Reference Lliboutry and VivetLliboutry and Vivet (1961). Measurements of the conductivity σ, by Reference Chaillou and VallonChaillou and Vallon (1964) showed also that in 1963 the plateau was covered by the upper layer of the 1962–63 winter snow accumulation, thickness 7 m. and quite low σ = 10⁻⁶ mho m.⁻¹, while just below this layer there was a thin layer of high σ ≈ 5×10⁻⁶ to 10⁻⁵ mho m.⁻¹, corresponding to the top surface of the glacier, with its accumulated dust and other impurities formed during the 1962 ablation season. At greater depths the glacial ice had a fifty times less in the upper layer. It should be noted that a variation of u with depth may distort an A.C. field penetration and distribution with a corresponding uncertainty in the depth being sampled by the A.C. field.

4.1 Greenland runs

Twelve useful runs were obtained on the snow of the “Big Array” glacier for six separations b in the range b = 1.8 m. to 190 m., using different lengths in the range l = 11 m. to 365 m. The Π-array was about 0.5 km. from the nearest rock outcrop and centred in a crevasse-free region of the glacier. The array was also moved from the smooth, level snow field to straddle some small surface cracks caused by thin crevasses under the surface, but there was no detectable change in f _m.

The maximum on the ϵ″−ϵ′ Cole–Cole diagrams was absent in all runs except one. This was due to the additional impurity loss factor ϵ‴ present for all frequencies below 6 kHz. A typical run is shown in the Cole–Cole diagram Figure 3. ϵ″ and ϵ′ are plotted in arbitrary units, since it is unnecessary here to know the absolute value of the geometrical factor G. The Cole–Cole semicircle for the array, b = 66.3 m., l = 110 m., has f _m = 5.0±0.2 kHz., and α = 0.64±0.05 radians. Further discussion of this and other “Big Array” Cole–Cole diagrams is deferred to Section 5.

Fig. 3. A Cole—Cole diagram for wet snow, measured on the “Big Array” glacier in West Greenland

Three runs were obtained for small b values on wet, bare glacial ice within 50 m. of a rock outcrop at the side of the glacier. A typical Cole–Cole diagram for the glacial ice is shown in Figure 4, in which two Cole–Cole semicircles can be constructed through the measured ϵ′ and ϵ″ values. The significance of the two semicircles is discussed in Section 4.2.

Fig. 4. A Cole—Cole diagram for glacial ice, measured on the “Big Array” glacier when the air temperature was below 0°C.

4.2 French runs

Runs were obtained in the Vallee Blanche for ten separations, in the range b = 1.8 m. to 170 m., and l = 8 m. to 140 m. Twenty runs, series O, were taken with the main arms of the array on the surface of the snow, while ten runs, series I, were taken with the array arms intentionally buried under a thin snow cover. Readings from run to run were less reproducible than in Section 4.1, while their scatter was greater, particularly at low frequencies and large ϵ‴.

The best series O runs shows two Cole-Cole semicircles. A typical run, with b = 136 m. and l = 91.5 m., is shown in Figure 5. The high-frequency semicircle had f _m = 2.7±0.1 kHz., and α = 0.58 radians, and is to be compared with the single Cole-Cole semicircles of the Section 4.1 runs. The lower frequency semicircle, with f _m ≈ 0.7±0.1 kHz., showed no spreading, α = 0, and it may represent the response of the snow or ice crystals close to, or frozen on, the surface of the wire electrodes, since it agrees with the response from a homogeneous sample without any spread in τ caused by temperature or density gradients in the sample.

Fig. 5. A Cole–Cole diagram for the névé of the Vallée Blanche, Francs. c₁. and c₂ are the centres of the two semicircles

The series I runs also gave two Cole-Cole semicircles and with less scatter in the readings. In general, the series I f _m values were only slightly lower than the series O f _m values, although J. G. Paren (private communication) has shown that C _s and f _m, should depend markedly on the proximity and degree of contact between the snow surface or crystals and the electrode surface.

4.3 Dependence of f_m and α on b

The f _m values, obtained from the runs of Sections 4.1 and 4.2, are plotted in Figure 6 as a function of the half-separation b/2. It has been considered, by analogy with the equivalent depth used in the determination of earth resistance, etc., that a given f _m value corresponds to a depth b/2 equal to one half of the electrode separation distance. Although this assumption may not be justified, it does not invalidate the discussion in Section 5 where the results of Section 4 have been interpreted in terms of the variation with depth of the physical properties of the snow and ice of a glacier. It can be seen that the shape of the f _m graph changed in going from (a) small depths less than about 13 m. where the two sets of results from the Sections 4.1 and 4.2 runs were quite different to (b) greater depths where the two sets were identical within the experimental uncertainty. At small depths b/2 < ~13 m., the “Big Array” f _m values decreased (Fig. 6, curve g) while the Vallée Blanche f _m values increased (Fig. 6, curve v), while at the greater depths f _m values were constant.

Fig. 6. The variation of the frequency f_m, at the mtlrirnum ϵ″ valve of the Cole—Cole semicircles, with the half-separation b/2. The solid-line error bars are the probable errors associated with two or more f _m values, the dashed error bars refer to single f _m values

The spreading factor a also varied with depth, as shown in Figure 7. α increased rapidly to a maximum value at a depth of about 3 m. and then decreased to a constant value at depths greater than about 10 m.

Fig. 7. The variation of the spreading factor a with the half-separation b/2 of the wire electrodes. Error bars are as in Figure 6

5.1 Model of a glacier studied by A.C. field

A cross-section of a glacier is illustrated schematically in Figure 8. A glacier is considered to consist of a series of layers of snow/ice, each layer with its different physical properties having a different relaxation time τ _s. Selected positions of the electrodes (i), (ii), with the associated electric force lines, radii b/2, are shown in Figure 8. The force lines were assumed to penetrate uniformly into the layers, and for the lines of radius b/2 to divide the total electric flux equally within and without the cylindrical surface defined by the electrode length l and the depth b/2. When the wires were in position (i), the relaxation time τ ₁ was that of layer 1, but when the wires were in position (ii), the measured τ _s was an average of τ ₁ and τ ₂ for layers 1 and 2. For the largest depths an average τ _s for all layers was measured. τ _s was not the arithmetical mean of τ ₁, τ ₂, etc., because for a distribution in τ _s the resultant Cole-Cole semicircle consisted of the envelope of the individual Cole-Cole semicircles of the layers, in which a slightly different Cole-Cole semicircle resulted from each layer with its different ϵ′ and ϵ″ values. This meant that a deeper layer, with its larger ϵ ₀ and ϵ″ max. values and hence Cole-Cole semicircle of larger radius, would tend to smear out the smaller semicircles of the shallower layers close to the surface, and the resultant τ _s would be more representative of the deeper layers than of the shallow layers.

Fig. 8. A hypothetical cross-section of a glacier illustrating the layer structure. The associated densities ρ, Formzahl u, and temperatures T at different depths are indicated. The semicircular field lines associated with the electrode positions (i), (ii), … are shown penetrating into layers 1, 2,…

5.2 Significance of Section 4.3

The important parameters of a layer are its temperature T, its density ρ _s and its form number u. The form number, or Formzahl, u expresses the relative proportions of the two components, air and ice, and their arrangement in the snow. τ _s is lengthened by an increase of ρ _s or u, and shortened by an increase in T, so that the τ _s value of a layer at a depth b/2 depends on the relative effects of ρ _s, u and T. During the summer ablation periods, a temperate glacier has a constant temperature T≃0°C., and any changes in τ _s with an increase in b/2 would be due to changes in ρ _s or u. On the other hand, during the winter accumulation period, when a positive temperature gradient exists downward into the glacier, the τ _s value depends on T as well as on ρ _s and u, and the changes in τ _s with b/2 would be due to the changes in ρ _s, u, and T. At large depths, all layers would be sampled by the A.C. field, and since T, ρ _s and u have reached constant values there would be no change in f _m or α. This was observed at larger depths, where in Figures 6 and 7 f _m and α were constant for b/2>~13 m.

On the model of a glacier, τ _s at constant temperature should increase, or f _m decrease, with an increase in depth. This was observed, see Figure 6, curve G, where at shallow depths f _m was of the order of 4 kHz., and decreased to f _m ≈ 3 kHz. at greater depths. The small-depth f _m ≈ 4 kHz. was an f _m value similar to that of wet glacial ice at 0°C. in the Athabasca Glacier, Columbia Icefield, Canada, measured with parallel plates placed in the surface ice by Reference Watt and MaxwellWatt and Maxwell (1960). At larger depths the present f _m value was lower than the Watt and Maxwell value, f _m ≈ 7 kHz., obtained with rod electrodes placed in a crevasse at a depth of about 10 m. The difference between the f _m values deep in the glacier may result from the differences either in the physical properties of the snow/ice samples, in the electrode arrays, or in the effect of ionic conductivity losses. Ionic conductivity losses appeared, on the basis of the Cole-Cole diagrams, (Fig. 3) to be larger in the “Big Array” glacier than in the Athabasca Glacier, which has σ = 5×10⁻⁷ mho m.⁻¹. Another difficulty in the interpretation of thef f _m values was that the difference between the f _m values for small and large depths was less than the difference to be expected from laboratory measurements (Yosida and others, 1958).

The variation of the spreading factor α at small depths b/2 was of interest. It might have been expected, that as b/2 increased, there would be an increase in the number of layers sampled until every layer was contributing to the measured ϵ′ and ϵ″ values, with the result that a would increase uniformly up to a constant value. Figure 7 shows that the runs, both at “Big Array” and Vallée Blanche, gave a maximum α value at b/2 ≈ 3 m. This 3 m. depth was small compared to for example the 7 m. depth of Vallée Blanche névé. However since a was obtained from the spread of the Cole-Cole semicircles, each corresponding to a different layer with its own ρ _s, u and T, there is no reason why a should not show a maximum value. This can be explained as follows. It is known that the ϵ ₀ and ϵ″ maximum values for snow are less than those of ice. At small b/2, a certain α was obtained from the envelope of the Cole-Cole semicircles for snow, but at greater b/2 these semicircles became swamped by the larger Cole-Cole semicircles of the higher ρ _s and u, consolidated snow of the lower layers of the glacier. An a value based on the larger semicircles would be smaller than one based on the intermediate situation where both small and large semicircles contributed to the measured α value, and a maximum α value would occur in the intermediate region. The relative importance of the small and large semicircles from different depths could not be determined directly, since the ϵ′ and ϵ″ values were arbitrary rather than absolute values.

It is also possible to propose an explanation of the low f _m, or long τ _s, values at small depths in the Vallée Blanche (Fig. 6, curve v). This result implied that the present method can detect temperature differences in a glacier, since τ _s, associated with the low temperature of the cold, dry Vallée Blanche névé, was longer than either τ _s of the deeper region of the Vallée Blanche glacier or τ of the wet, 0°C. “Big Array” snow. However, there was a fundamental difficulty in relating f _m to the correct temperature, since the temperature dependence of f _m is not known. Nevertheless, an estimate of the upper limit of temperature in the Vallée Blanche névé can be made, if three assumptions are used: (a) that at 0°C. f _m ≈ 3 kHz., (b) that f _m for snow has the logarithmic temperature dependence of Equation (1), and (c) that the activation energy of névé, W _n, is less than W ₁ of ice. The estimated temperature then is less than −5.0°C.: the correct temperature would have been in the range −15°C. to −25°C.

The depth of the 0°C. isotherm was about 13 m. down in the Vallée Blanche névé at the time of the series O runs. This followed from the constant value f _m ≈ 3 kHz, for snow at 0°C. at depths greater than about 13 m. A 13 m. depth was in agreement with a measured position of the 0°C. isotherm at about 15 m. just before the start of the summer ablation period (Reference LliboutryLliboutry, 1965).

The constant f _m value at the largest depths ~ 20 m. indicated that the bedrock had not affected the f _m value. No effect was expected for the Vallée Blanche where the bedrock was about 180 m. below the surface of the snow. The constancy of the f _m values on the “Big Array” glacier indicated that its depth was greater than about 100 m. Further work is planned to use the method for the determination of the depth of snow and ice bodies.

In summary, the dielectric relaxation measurements have indicated that

• i. temperature differences due to the penetration of a winter cold wave into névé can be detected,

• ii. the 0°C. isotherm in a temperate glacier can be located,

• iii. a detailed theory of the dielectric properties in a glacier is lacking.